MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , min(add(n, nil())) -> n , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) , if_min(true(), add(n, add(m, x))) -> min(add(n, x)) , if_min(false(), add(n, add(m, x))) -> min(add(m, x)) , head(add(n, x)) -> n , tail(nil()) -> nil() , tail(add(n, x)) -> x , null(nil()) -> true() , null(add(n, x)) -> false() , rm(n, nil()) -> nil() , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) , if_rm(true(), n, add(m, x)) -> rm(n, x) , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x)) , minsort(x) -> mins(x, nil(), nil()) , mins(x, y, z) -> if(null(x), x, y, z) , if(true(), x, y, z) -> z , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z) , if2(true(), x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil(), app(z, add(head(x), nil()))) , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) } Obligation: runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 60.0 seconds. 2) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'WithProblem (timeout of 30 seconds) (timeout of 60 seconds)' failed due to the following reason: Computation stopped due to timeout after 30.0 seconds. 2) 'Fastest (timeout of 5 seconds) (timeout of 60 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 3) 'Best' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS) (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 2) 'bsearch-popstar (timeout of 60 seconds)' failed due to the following reason: The processor is inapplicable, reason: Processor only applicable for innermost runtime complexity analysis 3) 'Innermost Weak Dependency Pairs (timeout of 60 seconds)' failed due to the following reason: We add the following weak dependency pairs: Strict DPs: { eq^#(0(), 0()) -> c_1() , eq^#(0(), s(x)) -> c_2() , eq^#(s(x), 0()) -> c_3() , eq^#(s(x), s(y)) -> c_4(eq^#(x, y)) , le^#(0(), y) -> c_5() , le^#(s(x), 0()) -> c_6() , le^#(s(x), s(y)) -> c_7(le^#(x, y)) , app^#(nil(), y) -> c_8(y) , app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , min^#(add(n, nil())) -> c_10(n) , min^#(add(n, add(m, x))) -> c_11(if_min^#(le(n, m), add(n, add(m, x)))) , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x))) , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x))) , head^#(add(n, x)) -> c_14(n) , tail^#(nil()) -> c_15() , tail^#(add(n, x)) -> c_16(x) , null^#(nil()) -> c_17() , null^#(add(n, x)) -> c_18() , rm^#(n, nil()) -> c_19() , rm^#(n, add(m, x)) -> c_20(if_rm^#(eq(n, m), n, add(m, x))) , if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x)) , if_rm^#(false(), n, add(m, x)) -> c_22(m, rm^#(n, x)) , minsort^#(x) -> c_23(mins^#(x, nil(), nil())) , mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z)) , if^#(true(), x, y, z) -> c_25(z) , if^#(false(), x, y, z) -> c_26(if2^#(eq(head(x), min(x)), x, y, z)) , if2^#(true(), x, y, z) -> c_27(mins^#(app(rm(head(x), tail(x)), y), nil(), app(z, add(head(x), nil())))) , if2^#(false(), x, y, z) -> c_28(mins^#(tail(x), add(head(x), y), z)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#(0(), 0()) -> c_1() , eq^#(0(), s(x)) -> c_2() , eq^#(s(x), 0()) -> c_3() , eq^#(s(x), s(y)) -> c_4(eq^#(x, y)) , le^#(0(), y) -> c_5() , le^#(s(x), 0()) -> c_6() , le^#(s(x), s(y)) -> c_7(le^#(x, y)) , app^#(nil(), y) -> c_8(y) , app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , min^#(add(n, nil())) -> c_10(n) , min^#(add(n, add(m, x))) -> c_11(if_min^#(le(n, m), add(n, add(m, x)))) , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x))) , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x))) , head^#(add(n, x)) -> c_14(n) , tail^#(nil()) -> c_15() , tail^#(add(n, x)) -> c_16(x) , null^#(nil()) -> c_17() , null^#(add(n, x)) -> c_18() , rm^#(n, nil()) -> c_19() , rm^#(n, add(m, x)) -> c_20(if_rm^#(eq(n, m), n, add(m, x))) , if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x)) , if_rm^#(false(), n, add(m, x)) -> c_22(m, rm^#(n, x)) , minsort^#(x) -> c_23(mins^#(x, nil(), nil())) , mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z)) , if^#(true(), x, y, z) -> c_25(z) , if^#(false(), x, y, z) -> c_26(if2^#(eq(head(x), min(x)), x, y, z)) , if2^#(true(), x, y, z) -> c_27(mins^#(app(rm(head(x), tail(x)), y), nil(), app(z, add(head(x), nil())))) , if2^#(false(), x, y, z) -> c_28(mins^#(tail(x), add(head(x), y), z)) } Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , min(add(n, nil())) -> n , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) , if_min(true(), add(n, add(m, x))) -> min(add(n, x)) , if_min(false(), add(n, add(m, x))) -> min(add(m, x)) , head(add(n, x)) -> n , tail(nil()) -> nil() , tail(add(n, x)) -> x , null(nil()) -> true() , null(add(n, x)) -> false() , rm(n, nil()) -> nil() , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) , if_rm(true(), n, add(m, x)) -> rm(n, x) , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x)) , minsort(x) -> mins(x, nil(), nil()) , mins(x, y, z) -> if(null(x), x, y, z) , if(true(), x, y, z) -> z , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z) , if2(true(), x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil(), app(z, add(head(x), nil()))) , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) } Obligation: runtime complexity Answer: MAYBE We estimate the number of application of {1,2,3,5,6,15,17,18,19} by applications of Pre({1,2,3,5,6,15,17,18,19}) = {4,7,8,9,10,14,16,21,22,25}. Here rules are labeled as follows: DPs: { 1: eq^#(0(), 0()) -> c_1() , 2: eq^#(0(), s(x)) -> c_2() , 3: eq^#(s(x), 0()) -> c_3() , 4: eq^#(s(x), s(y)) -> c_4(eq^#(x, y)) , 5: le^#(0(), y) -> c_5() , 6: le^#(s(x), 0()) -> c_6() , 7: le^#(s(x), s(y)) -> c_7(le^#(x, y)) , 8: app^#(nil(), y) -> c_8(y) , 9: app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , 10: min^#(add(n, nil())) -> c_10(n) , 11: min^#(add(n, add(m, x))) -> c_11(if_min^#(le(n, m), add(n, add(m, x)))) , 12: if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x))) , 13: if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x))) , 14: head^#(add(n, x)) -> c_14(n) , 15: tail^#(nil()) -> c_15() , 16: tail^#(add(n, x)) -> c_16(x) , 17: null^#(nil()) -> c_17() , 18: null^#(add(n, x)) -> c_18() , 19: rm^#(n, nil()) -> c_19() , 20: rm^#(n, add(m, x)) -> c_20(if_rm^#(eq(n, m), n, add(m, x))) , 21: if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x)) , 22: if_rm^#(false(), n, add(m, x)) -> c_22(m, rm^#(n, x)) , 23: minsort^#(x) -> c_23(mins^#(x, nil(), nil())) , 24: mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z)) , 25: if^#(true(), x, y, z) -> c_25(z) , 26: if^#(false(), x, y, z) -> c_26(if2^#(eq(head(x), min(x)), x, y, z)) , 27: if2^#(true(), x, y, z) -> c_27(mins^#(app(rm(head(x), tail(x)), y), nil(), app(z, add(head(x), nil())))) , 28: if2^#(false(), x, y, z) -> c_28(mins^#(tail(x), add(head(x), y), z)) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { eq^#(s(x), s(y)) -> c_4(eq^#(x, y)) , le^#(s(x), s(y)) -> c_7(le^#(x, y)) , app^#(nil(), y) -> c_8(y) , app^#(add(n, x), y) -> c_9(n, app^#(x, y)) , min^#(add(n, nil())) -> c_10(n) , min^#(add(n, add(m, x))) -> c_11(if_min^#(le(n, m), add(n, add(m, x)))) , if_min^#(true(), add(n, add(m, x))) -> c_12(min^#(add(n, x))) , if_min^#(false(), add(n, add(m, x))) -> c_13(min^#(add(m, x))) , head^#(add(n, x)) -> c_14(n) , tail^#(add(n, x)) -> c_16(x) , rm^#(n, add(m, x)) -> c_20(if_rm^#(eq(n, m), n, add(m, x))) , if_rm^#(true(), n, add(m, x)) -> c_21(rm^#(n, x)) , if_rm^#(false(), n, add(m, x)) -> c_22(m, rm^#(n, x)) , minsort^#(x) -> c_23(mins^#(x, nil(), nil())) , mins^#(x, y, z) -> c_24(if^#(null(x), x, y, z)) , if^#(true(), x, y, z) -> c_25(z) , if^#(false(), x, y, z) -> c_26(if2^#(eq(head(x), min(x)), x, y, z)) , if2^#(true(), x, y, z) -> c_27(mins^#(app(rm(head(x), tail(x)), y), nil(), app(z, add(head(x), nil())))) , if2^#(false(), x, y, z) -> c_28(mins^#(tail(x), add(head(x), y), z)) } Strict Trs: { eq(0(), 0()) -> true() , eq(0(), s(x)) -> false() , eq(s(x), 0()) -> false() , eq(s(x), s(y)) -> eq(x, y) , le(0(), y) -> true() , le(s(x), 0()) -> false() , le(s(x), s(y)) -> le(x, y) , app(nil(), y) -> y , app(add(n, x), y) -> add(n, app(x, y)) , min(add(n, nil())) -> n , min(add(n, add(m, x))) -> if_min(le(n, m), add(n, add(m, x))) , if_min(true(), add(n, add(m, x))) -> min(add(n, x)) , if_min(false(), add(n, add(m, x))) -> min(add(m, x)) , head(add(n, x)) -> n , tail(nil()) -> nil() , tail(add(n, x)) -> x , null(nil()) -> true() , null(add(n, x)) -> false() , rm(n, nil()) -> nil() , rm(n, add(m, x)) -> if_rm(eq(n, m), n, add(m, x)) , if_rm(true(), n, add(m, x)) -> rm(n, x) , if_rm(false(), n, add(m, x)) -> add(m, rm(n, x)) , minsort(x) -> mins(x, nil(), nil()) , mins(x, y, z) -> if(null(x), x, y, z) , if(true(), x, y, z) -> z , if(false(), x, y, z) -> if2(eq(head(x), min(x)), x, y, z) , if2(true(), x, y, z) -> mins(app(rm(head(x), tail(x)), y), nil(), app(z, add(head(x), nil()))) , if2(false(), x, y, z) -> mins(tail(x), add(head(x), y), z) } Weak DPs: { eq^#(0(), 0()) -> c_1() , eq^#(0(), s(x)) -> c_2() , eq^#(s(x), 0()) -> c_3() , le^#(0(), y) -> c_5() , le^#(s(x), 0()) -> c_6() , tail^#(nil()) -> c_15() , null^#(nil()) -> c_17() , null^#(add(n, x)) -> c_18() , rm^#(n, nil()) -> c_19() } Obligation: runtime complexity Answer: MAYBE Empty strict component of the problem is NOT empty. Arrrr..